B.8 For further reading
1.5.2 A two-stage decision process: Production planning in an assemble-to-order environment
IVhen computing safety stocks, we do not plan orders in advance: SVe pre- scribe the structure of a policy (e.g.. Q and R). and we let the system run and place orders when our policy suggests t o do so ( e g . , when we hit the reorder point R). In practice. the parameters are adjusted periodically. Furthermore, emergency actions are carried out when needed. All of these adjustments are carried out when additional information is obtained, but this is outside this formal model. The formal model is. in a sense. single-stage: SVe make some decisions and then see what happens. In some other cases, we want t o include in a formal model the adjustments we might make at a later stage. In order t o do so. we must formalize the dynamics of the decision process. whereby de- cisions are made and/or revised when new information is obtained. This may lead to very difficult stochastic models. Lye consider here a simple example of a two-stage model.
Consider an assemble-to-order (ATO) system. In such a system. we have to make (or buy) components. which are then assembled into some end item we sell. It would be nice to do everything after we receive a customer order.
but we cannot afford this luxury if the customer is not willing to wait t ha t much time. If the customer wants everything immediately. we have to keep a stock of end items; this may be difficult or impossible when end items come in a wide variety of configurations or when items cannot be stocked because of their cost. A compromise solution is feasible when making components requires a long lead time. but assembly is relatively fast. We can keep a stock
DEALING WITH UNCERTAINTY 31
of components. which are made or bought before we get customer orders. l y e assenible only on order. i.e.. after we collect customer demand. Concrete examples of AT0 processes are the automotive industry, at least, in Europe.
and the PC indust,ry, where one can order a customized model and select among a number of feature/opt,ions.26
Let us build a simple but instructive model along with a small numerical example. under the following assumptions:
1. First'. 1%-e decide how many units of each component we build. subject to manufacturing capacity constraints. This first decision sets the total production cost.
2 . After receiving cust,omer orders. we use components to assemble finished goods. The assembly plan in designed t o maximize revenues: the cost term in the profit function is fixed by the previous decision (if we neglect assembly cost): if components are not enough to meet cust,omer orders.
we lose profit opportunities: if too many components are available. they are discarded with a possibly considerable loss of money
The key- point, apart from demand uncertainty. is that we have a limited t,inie window for sales. after which components are no use. This is a limit assumption. typical of the classical newsvendor model (see section 5 . 2 ) : in practice. components might have some salvage value, or they could be used in later time periods. In this setting. we have t o make two decisions i i i sequence, in order to optimize profit. Literally. we cannot maximize profit. becausi: it is a random variable depending on our decisions and on uncertain demand. but 1%-e may maximize its expected value.27
Since t'he main complicating factor is demand uncertainty, one possibility is t o disregard it and just use expected values of demand in planning production of components. Another possibility is representing demand uncertainty by a set of scenarios. We will pursue both approaches and compare the decisions we make.
To set up a small toy example, say t h a t we own a (very) small firm. pro- ducing just three end items
(Al.
i l 2 . A3). which are obtained by assembling components (q, c2. c3. cq. cg). The components we use for each end item are described b>- a bill of materials, which is flat (just two levels: end items and components). The bill of mat,erials is given in the left-hand side of table 1.1.From the bill of materials, we see that there are two common components. c1 and c2, while the remaining tliree are specific and characterize each end it,ein.
15'e assume that three resources (AII1. :U2. -213) are used for production of coin- ponents. On the right-hand side of the table w e also see the bill of resourceb,
2 6 A possitil) more pleasing example is any pizzeria offering a wide array of pizzas: the. pizza is made on order. but all of the components are prepared in advance.
2 7 A more sophisticated approach would involve some considerations a b o u t risk. Lvhich is not fully captured by t h e expected value.
Table 1.1 Bill of materials for the assemble-to-order example
Table 1.2 Bill of resources. cost of components. and available capacity
A1 1 1 1 0 0
A 2 1 1 0 1 0
A 3 1 1 0 0 1
c1 1 2 1 2 0
c2 1 2 2 3 0
c3 2 2 0 1 0
c4 1 2 0 1 0
c5 3 2 0 1 0
Cap. 800 700 600
Table 1.3 Demand scenarios. expected value of demand. and selling price of end items
Sl
5’2 5’3 Exp. Demand Selling PriceA1 100 50 120 90 80
A2 50 25 60 45 70
A3 100 110 60 90 90
i.e.. the time required on each resource to manufacture one component. In the table, we also give the available capacity for each resource type, and the cost of each component: this cost might include both direct variable production costs and material costs. We assume that assembly is not a bottleneck, hence its capacity is disregarded.
Other relevant d a t a concern end items, demand, and the price at which end items are sold. They are given in table 1.3. Demand uncertainty is modeled by a set of three scenarios (S1, 5’2. S3). If we have information about past sales. the three scenarios may result from the discretization of a continuous probability distribution (of course, more scenarios are needed in a practical setting to approximate the distribution): alternatively, they could result from an interview with three experts. N’hatever the case, we assume that the three scenarios are equally likely. i.e.. each probability is 1/3.28 15-e also give the expected value of demand. which is obtained by averaging the three scenarios for each end item. The last column displays the price a t which end items are sold.” Also, note that the selling price is larger than 60, the total component
28When discretizing continuous distributions, we might use different probabilities t o get a better approximation: see. e.g.. [4. chapter 101 for an application of Gaussian quadrature. In t h e case of forecasts based on subjective judgment by experts, using the same probabilities means t h a t we consider three equally reliable experts.
2gIf we do not want t o disregard assembly cost, we may substitute selling price by contri- bution t o profit from assembling and selling an item: this defines t h e second-stage cost. as it takes selling price and assembly cost into account. but not component costs.
D f A L l N G WlTH UNCERTAlNTY 33
cost. for all of the three end items. but A3 looks more profitable. in a sense.
because its profit margin including component costs is 90 - GO = 30. uhereas
A2
is the least profitable: of course. this reasoning may be misleading in that it does not take into account resource c o n s ~ m p t i o n . ~ ~\Ye may tackle the problem of maximizing expected profit by the Linear Programming (LP) techniques described in appendix B. To build a simple model as a starting point. we could disregard uncertainty and deal with one scenario characterized by average demand. \Ye get the following model:
3 3
i = l ;=1
s.t.
2
Timzi 5 L,. m = 1 , 2 , 3 . i=lyJ 5 d j . j = 1 , 2 , 3 .
3
,=1
Y j , X Z
2
0.Here. subscript i refers to components: subscript j refers to end items; and subscript' m refers to resource types. Input data correspond to those reported in the tables:
0 the component cost Ci:
0 the selling price P, for each end item:
0 the available capacity L, for each resource type (measured in time units):
the resource requirement (processing time) Tzm: for component i on resource m:
0 the number G,; of components i going into an end item j (i.e.. the bill the expected demand
d,.
which is assumed certain.of materials - BOhl);
The decision variables are 2 , . the number of component,s of typt' i that we produce, and ,yJ. the number of end items of type j that are assembled arid sold; to be more precise. we pretend that we will really sell an miount y,.
because we disregard demand uncertainty. The model aims at maximizing profit. as expressed by the objective function (1.4). subject to capacity con- straints (1.5). The inequality (1.6) stat,es that we cannot sell more than what
" S e e example B . l on page 537
is demanded, whereas (1.7) says that we cannot assemble end items if the required components are not available. The decision variables are required to be non-negative. In fact, for the sake of simplicity, we consider a contznuous LP model, which allows for fractional quantities; if we insist on requiring t ha t produced and assembled quantities are integer, it is easy to incorporate this requirement (see section B.6).
Solving the model, e.g., by the simplex method (see appendix B). we get the following solution (rounded to two decimal digits):
ZT
= 116.67,X: = 26.67. xi = 0.00, xi = 90.00.
IJT
= 26.67. y; = 0.00. 1~: = 90.00.X; = 116.67,
In this very small example, we may easily interpret what this solution tries to accomplish. We assemble the maximum number of end items of type A3.
which is the most profitable one; this requires in turn the production of a corresponding number of common components c1 and c2. as well as the specific component c5. Since demand limit is binding for AS. there is some capacity left, which is used to produce a limited amount of the specific component
c3, which is needed to assemble end item Al, plus common components. End item A2 has the lowest selling price and is disregarded, as well as is its specific component c4. It should be noted t h at , in general. one should not take for granted that the production of the highest profit item should be maximized:
the consumption of available resources should be taken into account as well (see example B . l on page 537 for a counterexample).
In this specific case, the solution is quite readable, but it is a bit "extreme."
An expert planner would immediately see t h at it is a risky bet on high sales of the most profitable item. The optimal profit. according to this model. is 3233.33. but this is actually misleading. After planning production of com- ponents, we do not know the value of profit, but only its distribution (if we accept the validity of the demand scenarios). We cannot maximize optimal profit: what we can do is maximizing its expected value, and this requires a more sophisticated model th at takes demand scenarios into account:
5 3
max - C C t x 2 + - - p ($P3YI)
,
(1.8),=1 s=1
5
s.t.
CT,,,Z,
5 L,. m = 1 . 2 , ~ (1.9)2 = 1
<
ds 3 = 1. 2.3. S = 1 , 2 , 3 . (1.10)- 3' 3
C G , , $ 5 Z, z = 1.2.3. 4.5, s = 1 , 2 , 3 . (1.11)
2=1
y;,xz 2 0.
DfALlNG WlTH UNCERTAINTY 35
The big change in this model. with respect to the expected demand model [(1.4)-(1.7)]. is that demand uncertainty is taken into account explicitly. Here we consider demand d i for item in scenario s. Accordingly, the quantity as- seinbled is now represented by scenario-dependent decision variables y; : this is the amount of end item we assemble and sell. if and when scenario s is real- ized. Assembly decisions are not taken here and now. when we plan produc- tion of components, but they are contzngent plans. The scenario-independent variables 2 , are first-stage variables, whereas variables y; are second-stage variables. So now we implement the production plan (i.e., first stage deci- sions 2 , ) and develop a contingency plan for the assembly operations (i.e , second stage decisions y i ) . Only when demand is realized we choose among the Contingency plans ( y i ~ y:. Y:).~' We should carefully notice the diffeience between a inultiperiod model and a multistage model. i4'e illustrate examples of inultiperiod models in appendix B and in chapter 4. In such models. deci- sions will be implemented in later time periods, but they are all taken now, based on the currently available information. It is possible t o revise such decisions bv solving the model again according to a rolling horizon strategy.
but this is outside the scope of the model itself. In a multistage model. we do not commit t o one specific decision for the later stages; the decision that will actually be implemented depends on the realization of random variables.
and it will be fixed only when the relevant information will be available in the future. Sext-stage variables may also be used to "adjust" previous decisions.
given current contingencies. This interpretation explains why models such as the one above are called stochastic programming models with recourse.
Going into details of the model above. the objective function (1.8) consists of a first-stage (deterministic) term accounting for the cost of components.
along with a second-stage term, which is the expected revenue from selling end items (not including component cost): the expected value is computed by summing the revenues under the three possible decisions, times scenario prob- abilities T ' . The capacith constraint (1.9) is unchanged. because it pertains to first-stage only. The market constraint (1.10) is now scenario-dependent.
as it considers the stochastic demand d;. Finally. constraint (1.11) links the two stages. stating that assembly is constrained by component availability, for each end item and each scenario. Solving this model. we get the following solution:
XT
= 115.71, Z; = 115.71.31Notice t h a t this holds only when t h e three scenarios a r e actually t h e only three possible demand scenarios. In other cases. we can face a very large number of different scenarios (possibly a n infinite number of different scenarios). In this case. t h e three scenarios are only meant t o model demand uncertainty and make sure t h a t first stage decibions account for demand uncertainty. T h e realized demand might differ from all three scenarios. In this case, once demand is realized we simply have to write a second model for assembly decisions.
where we need t o meet t h e realized demand with a limited quantity of components t h a t was fixed through t h e above model.
The real outcome of the model is the set of the first-stage decision variables x,*. Observing the component production plan. we immediately see a quali- tative difference with respect t o the model disregarding uncertainty: It is less extreme. We do not produce a large amount of component c j , because we do not place a risky bet on high sales of A3. In fact, scenario three would prove a disaster for the deterministic solution: In that scenario, sales are lower for AS.
but we could not react because we do not have enough specific components for the other end items. This also implies th at many specific components32 would be thrown away (according t o our assumptions concerning the limited time window for sales and the lack of any salvage value of unused components).
The stochastic model. instead, increases production of specific component c3, which is needed to support assembly and sales of A l : even a small amount of component c4 is produced, in order to support the least profitable end item
A2, which helps in using common components when sales are low for other end items. While there is a big difference in terms of specific components, we see that as far as common components are concerned, the solutions of the deterministic and the stochastic solutions are essentially the same. There is a good reason for this. as common components are a flexible resource, which can be exploited to support different end items. Moreover, the demand for common components is the sum of the individual demands for the end items.
and by aggregating demand we often reduce uncertainty. Indeed, this rzsk poolsng effect is what we try t o exploit in assemble-to-order systems. In chap- ter 6 we will see t h at the same mechanism is exploited in the management of distribution networks. However. it is also important to note that when end item demands are strongly correlated. the risk pooling effect is considerably reduced. In such a case. we should expect that even the produced quantities of common components differ in the deterministic and the stochastic model.
Another relevant factor is capacity: If this is so tight that we may sell what- ever we are able to produce. a simple deterministic model could be a viable option.
But how do the two solutions compare in terms of profit? The objective function from the solution of the second model is 2885.71: apparently. the stochastic solution is worse t h an the deterministic solution. whose objective value was 3233.33. But this comparison makes no sense. We are actually comparing two different situations rather than two different solutions. The above finding simply proves t h at we would rather face a certain demand rather than a n uncertain one. The objective function of the first model is neither
321n t h e more general case even common components could be thrown away.
DEALING WITH UNCERTAINTY 37
the true profit. which is uncertain. nor its expected value. It would be the optimal profit. if we knew that the average demand scenario is mhat will be realized. In the first model I1.4)-(1.7)] we pretend t o know the end item demand. and we get the illusion of higher profits In order t o compare the t n o solutions, we should fix the production plans for components sugge5ted by the two models, and then we should solve a set of second-stage problems, where we optimize assembly of end items subject t o component availability, for different demand scenarios. Nore formally. given a set of first-stage variables 2 ; for components, we should solve the following second-stage (recourse>) problem for each scenario s'
C
3 Gzl$ 5 x,'.J = 1
>
0,YJ -
i = 1 . 2 , 3 . 4 . 5 ,
where
R"(x*)
is the optimal revenue we collect under scenario s. given the first-stage solution XI. and making optimal use of available components to meet demand. The first-stage solution can come from solving a stochastic or an expected-value model: whatever the case. its expected revenue isS
Expected profit for an arbitrary solution can be obtained by subtracting its first-stage cost from its second-stage expected To evaluate the deterministic solution. we should plug it in this model: in case of scenario S1.
the optimal assembly and sales plan is
y; = 26.67. y; = 0.00, yg = 90.00.
and the hame holds for Sz. The bad news is that if scenario 5'3 occurs. we are i n trouble. because the high-risk solution does not fit demand verv well The optimal assembly and sales plan would be
yr = 26.67. pE = 0.00, g: = 60.00.
This is a pretty bad scenario with low sales and corresponding low profit.
JYe must compute revenue for each scenario. multiply it by its probability.
33\fe are evaluating expected profit in-sample. i.e., by- using t h e same set of scenarios which are used in t h e stochastic model; we could use a much larger set of out-of-sample scenarios t o get a more reliable estimate. T h e point is t h a t solving a large number of small L P problems may take less C P U time t h a n solving one large stochastic LP model.
sum everything t o get the expected value, and subtract the component cost from the first stage. Doing so, we may see that the expected profit from the deterministic solution (2333.33) is much lower than what the objective function of the deterministic model [(1.4)-(1.7)] predicts (3233.33). based on one average-case scenario. The percentage improvement of the stochastic solution with respect t o the deterministic one is
2885.71 - 2333.33
=
23.67%.2333.33
Clearly, we cannot extrapolate general results from a small toy example.
Indeed, the advantage of using a stochastic model is striking here, because specific components have a large impact. In a case featuring a lot more com- ponent commonality, the result would be less impressive. Furthermore. we have assumed that unused components are scrapped, which need not be the case. They could have some salvage value, and we could have a multistage problem so that they can be used in later stages. Nevertheless. the example is quite instructive in pointing out:
0 the difference between decision stages and tame periods.
0 the role of risk pooling.
In this case. risk pooling is obtained by using common components and by deferring assembly decisions. To further illustrate the value of deferring de- cisions in a more specific distribution setting, some fashion retail chains send only a part of the items to retail stores a t the beginning of a season: at a later stage, after observing sales at each retail store. the residual stock is sent down- stream. Also in this case. the first decision, i.e., the purchase of items from suppliers, is often constrained by a budget assigned to each buyer in charge of a specific market segment. The second decision, inventory allocation, can be made by a different type of professional called planner.
A last important consideration, which applies to all models we describe in this book to deal with demand uncertainty. is that we have considered the maximization of expected profit as a suitable objective. We do not consider profit variability across scenarios. or what happens in extremely bad scenarios (the average smoothes out single outcomes). This makes sense if we may repeat the game over and over (for various items or over multiple periods).
so that what really matters is the average profit in the long run. However, in the short run we may take too many chances: If a single bad decision cannot be recovered. because we immediately go out of business, or get fired. a more careful approach should be taken to fully account for risks. An alternative view, for economically minded readers. is that considering expected profit is equivalent to assuming a risk-neutral attitude: risk-averse decision makers should consider different objective functions.